Simplify the following expression and state the condition under which the simplification is valid. You can assume that $t \neq 0$. $r = \dfrac{50t - 10}{2t} \div \dfrac{35t - 7}{9} $
Dividing by an expression is the same as multiplying by its inverse. $r = \dfrac{50t - 10}{2t} \times \dfrac{9}{35t - 7} $ When multiplying fractions, we multiply the numerators and the denominators. $r = \dfrac{ (50t - 10) \times 9 } { 2t \times (35t - 7) } $ $ r = \dfrac {9 \times 10(5t - 1)} {2t \times 7(5t - 1)} $ $ r = \dfrac{90(5t - 1)}{14t(5t - 1)} $ We can cancel the $5t - 1$ so long as $5t - 1 \neq 0$ Therefore $t \neq \dfrac{1}{5}$ $r = \dfrac{90 \cancel{(5t - 1})}{14t \cancel{(5t - 1)}} = \dfrac{90}{14t} = \dfrac{45}{7t} $